The preceding analysis drew some pretty pictures. It then suggested that the dynamics in those pictures cast doubt on the informativeness of standard growth regressions. But is there an economic structure that could produce such dynamics?
This section describes a model that does so.
It is standard in many growth analyses to study how one economy, in isolation from all other economies, grows quickly or slowly.12 The insights developed are then used to explain why some countries grow faster than others.
The point made in the previous section bears on this and, with hindsight, is obvious: studying an average or representative economy gives little insight into the empirical behavior of the entire cross section. For such cross section dynamics to be interpretable, one needs a theoretical model that makes predictions on them.
The model here draws from [30]. It makes predictions on cross section dynam-ics by linking three observations: (i) countries endogenously select themselves into groups, and thus do not act in isolation; (ii) specialization in production allows exploiting economies of scale; and (iii) ideas are an important engine of growth.
The key results are two: (a) coalitions|or, as it will turn out, convergence clubs|form endogenously; the model delivers predictions on coalition membership across the entire cross section of economies; (b) di erent convergence dynamics are generated, depending on the initial distribution of characteristics across countries.
Included among these potential dynamics are explicit convergence club character-izations; polarization|the rich becoming richer, the poor poorer, and the middle class vanishing; strati cation|multiple modes in the income distribution across countries; and overtaking and divergence|two economies initially on roughly equal footing, separating over time so that one eventually becomes wealthier than the other.
12 Exceptions do exist, e.g., [18, 21].
In [30] I show that coalitions, in equilibrium, are typically nondegenerate, nontrivial subsets of the entire cross section of economies. The con guration of coalitions comes from recognizing the forces across countries for consolidation and fragmentation. Equilibrium balances these opposing forces, and permits (at least for a time) diversity. Coalitions can then be recognized as \convergence clubs,"
thereby determining international patterns of growth, convergence, and polariza-tion.
The resulting dynamics are in general subtle, but special cases can be graph-ically represented. Figure 4.1 shows some equilibrium time paths for incomes (discounted by the steady-state growth rate).
In this example, at timet0 there is some initial income distribution across the cross section of economies. Over time, some economies become better o , others worse o ; overtaking is possible. Coalitions or convergence clubs form, however, and the distribution tends towards a bimodal distribution at time t1. In general, the number of modal points equals the number of coalitions that form. When more than two coalitions form, strati cation is an apposite term in place of polarization to describe the outcome. With two coalitions (as in gure 4.1), the distribution dynamics can be easily seen. Eventually, the middle-income group of economies vanish, and the rich continue to become richer, and the poor, poorer. Clustering occurs at high and low parts of the income distribution.
The exact outcome|the number of coalitions, their composition, and so on|
depends on the initial distribution of income across the entire cross section. If the world began with all incomes already close together, then only a single coalition forms; all countries then converge to equality. If, on the other hand, initial incomes are disparate, then, more likely, multiple convergence clubs form. The distribution dynamics will then be the multi-mode or strati cation extension of gure 4.1.
Economists observing these dynamics are naturally led to describe the resulting groups as convergence clubs.
But what happens if researchers apply the tools of -convergence analysis to data generated by gure 4.1? A researcher might attempt to understand the
behavior of incomes across the cross section of economies by, say, \controlling"
for di ering human capital stocks and other observable variables. That researcher could then conclude that conditional convergence occurs, and that human capital explains cross-country patterns of growth. However, such conclusions mislead.
It is, instead, the pattern of club membership that explains everything. In the model, human capital is only responding endogenously to coalition structures:
that is why high human capital is found among rich-club countries. Moreover, conditional convergence is not a useful way to think about the polarization induced by convergence-club formations: the interesting strati cation in gure 4.1 is never revealed by conditional-convergence investigations.
The dynamics in gure 4.1 would be one motivation for the empirics devel-oped in [11]. There, Durlauf and Johnson provide an innovative technique for consistently uncovering local basins of convergence|they do this by allowing their tted regression model to \adapt" subsamples, depending on the data realizations themselves. What I give below is a di erent empirical method, but one that seems to me more natural for studying evolving distributions. Durlauf and Johnson in-terpret their empirics in terms of multiple regimes. I inin-terpret gure 4.1 as just one equilibrium law of motion, but in an entire distribution, which could then have multiple modal points in the ergodic limit. In general, each technique|that in [11] and the section below|has advantages in di erent dimensions over the other;
neither strictly dominates.
Finally, the empirical message here is not con ned to this particular model of ideas and growth. In [33] I adapted the model in [15] to produce much the same empirical results as in gure 4.1. Other models employing local nonconvexities in the technology, nonlinearity in the savings function, and so on will, again, give the same empirics (see, e.g., [6]).